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Reference6 articles.
1. This was pointed out by G. Doetsch as early as 1923 and used by him and others since then in the solution of boundary-value problems. See for instance the series of five papers by G. Doetsch, Probleme aus der Theorie der Wärmeleitung: I (with F. Bernstein). Eine neue Methode zur Integration partieller Differentialgleichungen. Der lineare Wärmeleiter mit verschwindender Anfangstemperatur, Math. Zeitschr.22 (1925), S. 285–292; II. Der lineare Wärmeleiter mit verschwindender Anfangstemperatur. Die allgemeinste Lösung und die Frage der Eindeutigkeit, Math. Zeitschr.22 (1925), S. 293–306; III. Der lineare Wärmeleiter mit beliebiger Anfangstemperatur. Die zeitliche Fortsetzung des Wärmezustandes, Math. Zeitschr.25 (1926), S. 608–626; IV (with F. Bernstein). Die räumliche Fortsetzung des Temperaturablaufs. Bolometerproblem, Math. Zeitschr.26 (1927), S. 89–98; V. Explizite Lösung des Bolometerproblems, Math. Zeitschr.28 (1928), S. 567–578. For an application of the method to a problem with variable coefficients see A. N. Lowan, On the cooling of a radioactive sphere. Physical Review44 (1933), pp. 769–775. Further applications are to be found in some of the references given below.
2. A discussion of some of these conditions is given by G. Doetsch, Les équations aux dérivées partielles du type parabolique, L'Enseignement Mathématique35 (1936), pp. 43–87.
3. R. V. Churchill, Temperature distribution in a slab of two layers, Duke Math. Journ.2 (1936), pp. 405–414; here the process used to obtain the complex integral form is too indirect to be generally useful. For conditions on the coefficients in a somewhat general problem, under which the solution can be written as a contour integral, see W. Mächler, Laplace'sche Integraltransformation und Integration partieller Differentialgleichungen vom hyperbolischen und parabolischen Typus. Commentarii Mathematici Helvetici5 (1933), S. 256–304. These conditions are quite narrow; in fact they are not satisfied by the example of displacements in a bar, given by the author in the last paper cited under 5), when the external force is constant.
4. T. von Stachó, Operationskalkül von Heaviside und Laplacesche Transformation, Acta Litt. ac Scient. Szeged3 (1927), S. 107–120; and G. Doetsch, Die Anwendung von Funktionaltransformationen in der Theorie der Differentialgleichungen und die symbolische Methode (Operatorenkalkül), Jahresber. D. M. V.43 (1934), S. 238–251.
5. Conditions which are similar to those in Theorem C, but broader, are stated by W. v. Koppenfels, Der Faltungssatz und seine Anwendung bei der Integration linearer Differentialgleichungen mit konstanten Koeffizienten, Math. Annalen105 (1931), S. 694–706 [S. 699], to be sufficient in order that 599-1; but his proof of their sufficiency seems to be faulty.
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